f(x) = (5x^3 + 1)^8 (4x^5 + 3)^7
How do I apply the chain rule to a problem like this?
Assuming you want to differentiate it, read it as a two-part product, apply the product rule, and each time you do a differentiation for the sake of the product rule you'll need to apply the chain rule - because each of the two parts of the product is a composite. Will put a pic, in a minute.
Just in case a picture helps...
... where (key in first spoiler)...
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Alternatively, take logs of both sides, then differentiate...
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Then solve the bottom row for f'(x).
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Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!
$\displaystyle u=5x^3+1,\ s=u^8=f(u)$
"u" is a function of x, "s" is a function of u.
$\displaystyle v=4x^5+3,\ t=v^7=f(v)$
"v" is a function of x, "t" is a function of v.
$\displaystyle f(x)=st$
$\displaystyle f'(x)=t\frac{ds}{dx}+s\frac{dt}{dx}$
But $\displaystyle s=f(u)$ and $\displaystyle t=f(v)$
hence, applying the Chain Rule...
$\displaystyle f'(x)=t\frac{ds}{du}\frac{du}{dx}+s\frac{dt}{dv}\f rac{dv}{dx}$